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    FMCW雷达距离多普勒(RDM)处理方法中距离分辨率和速度分辨率的推导 |  GtJason1998
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  FMCW雷达距离多普勒(RDM)处理方法中距离分辨率和速度分辨率的推导
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      <h1 id="1-距离多普勒-Range-Dopple-Matrix-处理方法"><a href="#1-距离多普勒-Range-Dopple-Matrix-处理方法" class="headerlink" title="1. 距离多普勒(Range-Dopple Matrix)处理方法"></a>1. 距离多普勒(Range-Dopple Matrix)处理方法</h1><p>&emsp;&emsp;众所周知，距离多普勒处理方法(Range-Dopple Matrix，简称RDM)是FMCW雷达进行多目标信息提取的有效手段，通过对雷达发送的多个周期的Chirp序列以及回波信息进行快时间维度和慢时间维度的处理，即可得到距离多普勒热力图，进而可以提取多目标的距离和速度信息。<br><a id="more"></a><br><img src="https://img-blog.csdnimg.cn/20200212132950250.jpg?x-oss-process=image/watermark,type_ZmFuZ3poZW5naGVpdGk,shadow_10,text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3FxXzQxMjQ4NDcx,size_16,color_FFFFFF,t_70#pic_center" alt="FMCW Multi-Chirp"><br>在FMCW的差拍信号中，我们知道，差拍信号的频率为</p>
<script type="math/tex; mode=display">f_{movingBeat} = f_{staticBeat} \pm f_d = \frac{2f_cR}{Ct_c} \pm \frac{2fv}{C} \tag 1</script><p>其中$f_{movingBeat}$和$f_{staticBeat}$分别为目标运动和静止状态下差拍信号的频率，$f_d$为多普勒频率，$f_c$为扫频带宽，$R$为目标距离，$C$为光速，$t_c$为扫频周期，$f$为Chirp信号中心频率，$v$为目标速度。</p>
<h2 id="1-1-快时间维度处理-Range-FFT"><a href="#1-1-快时间维度处理-Range-FFT" class="headerlink" title="1.1. 快时间维度处理(Range-FFT)"></a>1.1. 快时间维度处理(Range-FFT)</h2><p>&emsp;&emsp;快时间维度即单个周期的Chirp序列扫频周期时间很短，短到几乎可以将多普勒频率带来的影响忽略不计($t_c$↓，公式(1)中$f_{staticBeat}$项占了主要的位置)，认为此时通过RDM热力图提取到的动目标在距离维度上的动目标差频$f_{movingBeat}$与静目标差频$f_{staticBeat}$近似相等，即</p>
<script type="math/tex; mode=display">f_{movingBeat} \approx f_{staticBeat} =  \frac{2f_cR}{Ct_c} \tag 2</script><p>那么通过快时间维度的每一帧数据，提取频谱峰值对应的横坐标频率，即可对目标的距离进行求解；即</p>
<script type="math/tex; mode=display">R = \frac{Ct_c}{2f_c}\cdot f_{staticBeat} \tag 3</script><p>快时间维处理示意图如下<br><img src="https://img-blog.csdnimg.cn/20200212133123832.jpg?x-oss-process=image/watermark,type_ZmFuZ3poZW5naGVpdGk,shadow_10,text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3FxXzQxMjQ4NDcx,size_16,color_FFFFFF,t_70#pic_center" alt="Range-FFT"></p>
<h2 id="1-2-慢时间维度处理-Doppler-FFT"><a href="#1-2-慢时间维度处理-Doppler-FFT" class="headerlink" title="1.2. 慢时间维度处理(Doppler-FFT)"></a>1.2. 慢时间维度处理(Doppler-FFT)</h2><p>&emsp;&emsp;因为我们知道，在快时间维的处理中，认为速度带来的影响忽略不计，通过对多个Chirp序列进行多帧数据的堆积，此时在第二个维度上(即慢时间维度上，多帧数据对应的同一距离单元上)速度带来的频率影响就不可忽略，此时慢时间维度上求得的频率即为多普勒频率，即</p>
<script type="math/tex; mode=display">f_d = \frac{2fv}{C} \tag 4</script><p>所以有</p>
<script type="math/tex; mode=display">v = \frac{f_dC}{2f} \tag 5</script><p>慢时间维处理示意图如下<br><img src="https://img-blog.csdnimg.cn/20200212134519777.jpg?x-oss-process=image/watermark,type_ZmFuZ3poZW5naGVpdGk,shadow_10,text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3FxXzQxMjQ4NDcx,size_16,color_FFFFFF,t_70#pic_center" alt="Doppler-FFT"><br>慢时间维度的处理是经过多个Chirp序列积累后对<strong>同一距离单元</strong>进行FFT的结果，故称为慢时间维度，</p>
<blockquote>
<p>为什么是同一距离单元？<br>因为Range-FFT中同一个横坐标对应相同的$f_{movingBeat}$，快时间维度下$f_{movingBeat}$约等于$f_{staticBeat}$，由公式(2)和公式(3)可知，对应同一距离单元</p>
</blockquote>
<p>快时间维度和慢时间维度处理总览<br><img src="https://img-blog.csdnimg.cn/20200212123652486.jpg?x-oss-process=image/watermark,type_ZmFuZ3poZW5naGVpdGk,shadow_10,text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3FxXzQxMjQ4NDcx,size_16,color_FFFFFF,t_70#pic_center" alt="两个维度总览"><br>经过处理后可得到如下的距离多普勒热力图(Range-Dopple Heat Map)<br><img src="https://img-blog.csdnimg.cn/2020021213115734.jpg?x-oss-process=image/watermark,type_ZmFuZ3poZW5naGVpdGk,shadow_10,text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3FxXzQxMjQ4NDcx,size_16,color_FFFFFF,t_70#pic_center" alt="Range-Dopple Heat Map"></p>
<h1 id="2-RDM中距离分辨率和速度分辨率推导方法"><a href="#2-RDM中距离分辨率和速度分辨率推导方法" class="headerlink" title="2. RDM中距离分辨率和速度分辨率推导方法"></a>2. RDM中距离分辨率和速度分辨率推导方法</h1><p>&emsp;&emsp;网上关于RDM方法中距离分辨率和速度分辨率推导的资料实在太少，几乎都是两个长得不太好看公式直接糊你脸上，我的感受就是<del>老人、地铁、看手机.jpg</del>(此处省略表情包)，于是决定记录下推导过程，正所谓难者不会，会者不难。<br><br>首先回顾下数字信号处理中第K个采样点的频率$f_k$与采样频率$f_s$间的关系，我们知道，第K个采样点的角频率服从如下关系</p>
<script type="math/tex; mode=display">\omega_k = \frac{k}{N_s}\cdot2\pi = \Omega\cdot T_s = 2 \pi f_k\cdot \frac{1}{f_s}</script><p>&emsp;其中$N_s$为采样点数，$\Omega$为模拟角频率，$T_s$为采样频率，我们取出等式中的第二项和第四项，有</p>
<script type="math/tex; mode=display">\frac{k}{N_s}\cdot2\pi =2 \pi f_k\cdot \frac{1}{f_s}</script><p>&emsp;可得第K个采样点的频率$f_k$与采样频率$f_s$间的关系为</p>
<script type="math/tex; mode=display">f_k = k\cdot \frac{f_s}{N_s} \tag 6</script><p>&emsp;到此就可以正式展开距离分辨率和速度分辨率的推导方法了，上一部分我们说到快时间维度的Range-FFT和慢时间维度的Doppler-FFT，有两个结论性的公式</p>
<script type="math/tex; mode=display">f_{movingBeat} \approx f_{staticBeat} =  \frac{2f_cR}{Ct_c} \tag 7</script><script type="math/tex; mode=display">f_d = \frac{2fv}{C}  \tag 8</script><p>&emsp;假设上一部分中距离多普勒热力图中，$n_1$为Range-FFT(快时间维度)中目标对应的坐标序列号，$n_2$为Doppler-FFT(慢时间维度)中同一目标对应的坐标序列号，则依照公式(6)可得</p>
<script type="math/tex; mode=display">f_{movingBeat} = \frac{n_1}{N_s} \cdot f_s \tag 9</script><script type="math/tex; mode=display">f_d = \frac{n_2}{N_{Chirp}} \cdot \frac{1}{t_c} \tag {10}</script><p>&emsp;其中$N_{Chirp}$为慢时间维度处理中Chirp序列的积累个数；公式(9)类比公式(6)，比较好理解，公式(10)也是类比公式(6)，只不过此时<strong>在慢时间维度上</strong>采样总数是积累的Chirp序列的总数，采样频率是每一个Chirp序列扫频周期的倒数，即$\frac{1}{t_c}$<br>&emsp;由此以来，分别联立公式(7)和公式(9)，联立公式(8)和公式(10)，可得</p>
<script type="math/tex; mode=display">\frac{2f_cR}{Ct_c} = \frac{n_1}{N_s} \cdot f_s</script><script type="math/tex; mode=display">\frac{2fv}{C} = \frac{n_2}{N_{Chirp}} \cdot \frac{1}{t_c}</script><p>&emsp;可解得</p>
<script type="math/tex; mode=display">R = \frac{C}{2f_C}\cdot t_c\cdot \frac{n_1}{N_s}\cdot f_s \tag{11}</script><script type="math/tex; mode=display">v = \frac{C}{2f} \cdot \frac{n_2}{N_{Chirp}}\cdot \frac{1}{t_c}\tag{12}</script><p>&emsp;因为 </p>
<script type="math/tex; mode=display">t_c = N_s\cdot T_s = \frac{N_s}{f_s} \tag{13}</script><script type="math/tex; mode=display">t_{seq} = N_{Chirp}\cdot t_c\tag{14}</script><p>&emsp;将(13)代入(11)，将(14)代入(12)，可得</p>
<script type="math/tex; mode=display">R = \frac{C}{2f_c} \cdot n_1</script><script type="math/tex; mode=display">v = \frac{C}{2ft_{seq}} \cdot n_2</script><p>&emsp;此时，就得到了距离分辨率和速度分辨率，分别为</p>
<script type="math/tex; mode=display">R_{res} = \frac{C}{2f_c}</script><script type="math/tex; mode=display">v_{res} = \frac{C}{2fN_{Chirp}t_c} = \frac{C}{2ft_{seq}}</script><h1 id="3-参考资料"><a href="#3-参考资料" class="headerlink" title="3. 参考资料"></a>3. 参考资料</h1><ul>
<li><a href="https://zhuanlan.zhihu.com/p/31047473" target="_blank" rel="noopener">Automotive radar 信号处理 第2课 速度估计</a></li>
<li><a href="https://zhuanlan.zhihu.com/p/77500626" target="_blank" rel="noopener">Radar测距及测速原理(2)——快速Chirp序列方法推导及实际应用</a></li>
<li><a href="https://mp.weixin.qq.com/s/3pmABLMsirzEXcIn3a2_Pw" target="_blank" rel="noopener">2D CFAR的原理，其实没那么的神奇</a></li>
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            Don't modify these 3 pswp__item elements, data is added later on. -->
        <div class="pswp__container">
            <div class="pswp__item"></div>
            <div class="pswp__item"></div>
            <div class="pswp__item"></div>
        </div>

        <!-- Default (PhotoSwipeUI_Default) interface on top of sliding area. Can be changed. -->
        <div class="pswp__ui pswp__ui--hidden">

            <div class="pswp__top-bar">

                <!--  Controls are self-explanatory. Order can be changed. -->

                <div class="pswp__counter"></div>

                <button class="pswp__button pswp__button--close" title="Close (Esc)"></button>

                <button class="pswp__button pswp__button--share" style="display:none" title="Share"></button>

                <button class="pswp__button pswp__button--fs" title="Toggle fullscreen"></button>

                <button class="pswp__button pswp__button--zoom" title="Zoom in/out"></button>

                <!-- Preloader demo http://codepen.io/dimsemenov/pen/yyBWoR -->
                <!-- element will get class pswp__preloader--active when preloader is running -->
                <div class="pswp__preloader">
                    <div class="pswp__preloader__icn">
                        <div class="pswp__preloader__cut">
                            <div class="pswp__preloader__donut"></div>
                        </div>
                    </div>
                </div>
            </div>

            <div class="pswp__share-modal pswp__share-modal--hidden pswp__single-tap">
                <div class="pswp__share-tooltip"></div>
            </div>

            <button class="pswp__button pswp__button--arrow--left" title="Previous (arrow left)">
            </button>

            <button class="pswp__button pswp__button--arrow--right" title="Next (arrow right)">
            </button>

            <div class="pswp__caption">
                <div class="pswp__caption__center"></div>
            </div>

        </div>

    </div>

</div>

<link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/photoswipe@4.1.3/dist/photoswipe.min.css">
<link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/photoswipe@4.1.3/dist/default-skin/default-skin.css">
<script src="https://cdn.jsdelivr.net/npm/photoswipe@4.1.3/dist/photoswipe.min.js"></script>
<script src="https://cdn.jsdelivr.net/npm/photoswipe@4.1.3/dist/photoswipe-ui-default.min.js"></script>

<script>
    function viewer_init() {
        let pswpElement = document.querySelectorAll('.pswp')[0];
        let $imgArr = document.querySelectorAll(('.article-entry img:not(.reward-img)'))

        $imgArr.forEach(($em, i) => {
            $em.onclick = () => {
                // slider展开状态
                // todo: 这样不好，后面改成状态
                if (document.querySelector('.left-col.show')) return
                let items = []
                $imgArr.forEach(($em2, i2) => {
                    let img = $em2.getAttribute('data-idx', i2)
                    let src = $em2.getAttribute('data-target') || $em2.getAttribute('src')
                    let title = $em2.getAttribute('alt')
                    // 获得原图尺寸
                    const image = new Image()
                    image.src = src
                    items.push({
                        src: src,
                        w: image.width || $em2.width,
                        h: image.height || $em2.height,
                        title: title
                    })
                })
                var gallery = new PhotoSwipe(pswpElement, PhotoSwipeUI_Default, items, {
                    index: parseInt(i)
                });
                gallery.init()
            }
        })
    }
    viewer_init()
</script>



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<script type="text/javascript" src="https://js.users.51.la/20544303.js"></script>

    
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